Hi! Please help me figure out up to 3 derivatives of: 25arctan(x)!!! I need the 1st, 2nd, and 3rd derivatives of this function!! Thank you so much!!!!
Hmm I'm kind of confused what you put. The original problem is to find the Taylor polynomial Tn(x) for the function f at the number a of f(x)=25arctan(x) when a=1 and n=3. I know how to do Taylor series, but I am having trouble finding the second and third derivatives to plug in 1 for them. I know that like for T2, it would be f(a)+f1st deriv. of a *(x-a) + f2ndderiv. of a over 2 factorial (x-a)^2. I just need help finding the 1st, 2nd, and 3rd derivatives. Thanks!!
haha Thank you! Well I know that you can pull out a constant for the normal derivatives that we did, but for Taylor series, do you do like (say for the 2nd derivative, the f(a) part) put 25f(a) of the 2nd deriv.? So would you evaluate the derivative you helped me with at a=1 and then whatever that equals, multiply it by 25? for all the derivatives? I know this is a dumb question but I think I'm confusing myself!
To get the 2nd and 3rd derivatives, you will need the quotient rule, which states that if $\displaystyle h(x) = \frac{g(x)}{f(x)}$ then $\displaystyle h'(x) = \frac{f(x)g'(x)-f'(x)g(x)}{f(x)^2}$
You could also (in my opinion this is preferable) use the chain rule and product rule. The product rule is $\displaystyle \frac{d}{dx}(uv) = u\frac{dv}{dx}+v\frac{du}{dx}$. Just stating the chain rule isn't very helpful for you, it really needs worked examples for you to understand it. You can find some of these here although if you have a good textbook it may well be easier to understand.